Throughout this note, X , Y, E and F will denote real BANACH spaces. The unit ball (sphere) of the BANACH space X will be denoted by B,(S,), and the term operator will mean a bounded linear function. An operator T: X + Y is said to be weakly precompact (wpc) if T(B,) is weakly precompact, i.e., (T(x,)) has a weakly CAUCHY subsequence for each sequence (x,) from B,. It follows easily from ROSENTHAL' S fundamental ['-theorem [R] that an operator T: X + Y is wpc if and only if TI, is not an isomorphism for any subspace Z of X isomorphic to e'. In this paper we will be concerned with studying operators T: X + Ysuch that the continuous adjoints T* : Y* + X * are wpc. The specific motivation for our results come from BOMBAL and PORRAS [BPI, ABBOTT, BATOR, BILYEU, and LEWIS [ABBL], ABBOTT, BATOR, and LEWIS [ABL], and BOMBAL and CEMBRANOS [BC].
Before stating our first result, we establish some additional notation and terminology. If T : X + Yis an operator, then a sequence (y:) in Y* is said to be T-weak*-null if (T(x), y,*) -+0 for each x EX. The sequence (en) will denote the canonical unit vector basis of c, , , and (e:) will denote the unit vector basis of 4'. If each of (x,) and (y,) is a basic sequence, we write (x,) -(y,) to signify that they are equivalent, and we denote the closed linear span of a subset A of X by [A]. The reader may consult DIESTEL [D] or [ABL] for any unexplained notation.
Our first result establishes a connection between wpc adjoints and strict singularity.